THE 2-LIEN OF A 2-GERBE

Transcription

1 THE 2-LIEN OF A 2-GERBE By PRABHU VENKATARAMAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2 c 2008 Prabhu Venkataraman 2

3 To my parents. 3

4 ACKNOWLEDGMENTS I would like to thank my advisor, Richard Crew, for guiding me through the literature, for suggesting this problem, and for patiently answering my questions over the years. I have learned a great deal from him. I am also grateful to David Groisser and Paul Robinson. Both of them helped me many times and served on my committee. Thanks also go to committee members Peter Sin and Bernard Whiting for their feedback on this project. The support of my family and friends has been invaluable to me. I thank all of them. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ABSTRACT CHAPTER 1 INTRODUCTION GERBES, LIENS AND 2-GERBES Torsors and H Fibered Categories and Stacks Gerbes and their Liens Categories, 2-Functors, 2-Natural Transformations Fibered 2-Categories, 2-Stacks and 2-Gerbes Representable Functors Representability Giraud s approach to Liens of Gerbes EQUALIZERS AND COEQUALIZERS Equalizers Representability of Equalizers in CAT Coequalizers Representability of Coequalizers in CAT GROUP CATEGORIES Inner Equivalences of Group Categories Action of a Group Category on a Category COEQUALIZERS AND QUOTIENTS IN ST ACKS Action of a Group Stack on a Stack Coequalizers in ST ACKS THE 2-LIEN OF A 2-GERBE Definition of a 2-Lien The 2-Lien of a 2-Gerbe Cocycle Description of the 2-Lien of a 2-Gerbe LIENS, PICARD STACKS AND H Gerbes and 3-Cocycles Liens and Strict Picard Stacks Strict Picard Stacks and H

6 8 CONCLUSION REFERENCES BIOGRAPHICAL SKETCH

7 Chair: Richard Crew Major: Mathematics Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE 2-LIEN OF A 2-GERBE By Prabhu Venkataraman May 2008 Principal bundles have a well-known description in terms of nonabelian cocycles of degree 1 with values in a sheaf. A more general notion than that of a sheaf on a space X is that of a lien on X. A lien on X is an object that is locally defined by a sheaf of groups, with descent data given up to inner conjugation. Equivalences classes of gerbes with a given lien L are classified by nonabelian degree 2 cocycles. In his work, Lawrence Breen has given a similar classification of 2-gerbes using nonabelian degree 3 cocycles that take values in a family of group stacks. In our work, we defined the notion of a 2-lien on a space X. It is an object that is given locally by a group stack, with 2-descent given up to inner equivalence. We have proved some theorems about 2-liens of 2-gerbes which correspond to well known results about liens of gerbes. Also, Deligne has shown that any strict Picard stack G corresponds to a 2-term complex of abelian sheaves K = [ K 0 d K 1 ]. In this case we proved that Ȟ3 (X, G) is isomorphic to the hypercohomology group Ȟ3 (X, K ). 7

8 CHAPTER 1 INTRODUCTION In geometry a fundamental concept is the notion of a principal G-bundle (also known as a G-torsor). If G is a sheaf on a space X, a G-torsor on X is a geometric realization of G-valued Čech 1-cocycle. The problem of defining non-abelian degree 2 sheaf cohomology leads to the concept of a gerbe. A gerbe on X may be thought of naively as a sheaf of categories with certain gluing axioms for objects and arrows. A gerbe on X is a geometric realization of a 2-cocycle on X with values in a non-abelian sheaf (or more generally a lien on X). This theory was worked out by Giraud [15]. Brylinski [7] developed a theory of differential geometry for gerbes. By his theory one can realize classes in H 3 (M, Z) as equivalence classes of abelian gerbes via the exponential map H 2 (M, C ) H 3 (M, Z). Then Murray invented the notion of a bundle gerbe [21]. The theory of bundle gerbes has proven very useful to physicists; for example bundle gerbes have been used to explore anomalies in quantum field theory [1], [2]. Hitchin has used the theory of gerbes for studying mirror symmetry [17]. Brylinski has used gerbes to give an interpretation of Beilinson s regulator maps in algebraic K-theory [8]. One can then ask, what kinds of objects are classified by non-abelian degree 3 sheaf cohomology? The answer is a notion which is built up from 2-categories which is due to Breen. He calls such objects 2-gerbes on X [4]. Brylinski and McLaughlin have studied a certain class of 2-gerbes [9], [10], namely 2-gerbes bound by the abelian sheaf C, which give rise to classes in H 4 (M, Z) via the exponential map H 3 (M, C ) H 4 (M, Z). They also prove that the class of the canonical 2-gerbe associated to a principal bundle P in H 4 (M, Z) is the same as the first Pontryagin class of P. This provides the motivation for our project. Given a 2-gerbe on X the associated 3-cocycle takes its values in a family of group stacks G i. Breen then points out that one could introduce the appropriately defined notion of a 2-lien L on a space X. Then the 8

9 3-cocycle could be viewed as defining a class in a corresponding L-valued cohomology set H 3 (X, L). In this thesis we define the notion of a 2-lien on a space X and show that each 2-gerbe gives rise to such a 2-lien. After proving some analogues of theorems about liens of gerbes, we establish some results about 2-liens that are strict Picard stacks, and an associated degree 3 hypercohomology group. We now provide a more detailed description of the contents of this thesis. In chapter 2, we review the necessary background material that forms the basis of this area, namely nonabelian cohomology. We begin with the definition of a G-torsor for a sheaf of groups G and explain their classification by H 1 (X, G). We then describe the objects that embody degree 2 cohomology: that is, gerbes. To do so, we first give the definition of a stack, a morphism between stacks and the definition of a 2-arrow between such morphisms. This data gives us a 2-category of stacks on a space X. After this setup, we give the definition of a gerbe and that of its associated lien. We then provide a description of the objects that embody degree 3 cohomology: that is, 2-gerbes. As with the case for gerbes, we first review the notion of 2-stacks prior to defining a 2-gerbe. The latter half of Chapter 2 is devoted to reviewing two notions from category theory that will prove essential to defining the notion of a quotient in the 2-category of stacks: these notions are those of the representability of a functor and of a 2-functor. We close by reviewing the original approach taken by Giraud [15] to introduce gerbes. It is this approach that we will adapt in order to define a 2-lien of a 2-gerbe in chapter 6. The key to defining a 2-lien is to figure out how to define a quotient in the 2-category of stacks. To answer this question locally, we first need to define a quotient in the 2-category of (small) categories CAT. This is what is accomplished in chapter 3: we prove the existence of a coequalizer in the 2-category of (small) categories CAT. Since the definition of an equalizer is used to define a coequalizer, we first prove the representability of an equalizer in CAT. We then construct a category Coeq corresponding to a (small) 9

10 pair of functors, and prove that this category represents the coequalizer of this pair in CAT. In chapter 4, we begin by giving a definition of inner equivalence for group categories and show that such inner equivalences for a fixed group category form a categorical group. We then define what it means for a group category to act on a category, and define the quotient by such an action using the concept of a coequalizer developed in chapter 3. In chapter 5, we begin by defining what it means for a group stack to act on a stack. Since the goal is to define the quotient by such an action, we do the work of proving that coequalizers are representable in the 2-category ST ACKS. This is done using the coequalizer construction in chapter 3, and via the universal properties of the coequalizer. After proving the existence of a coequalizer in ST ACKS we are able to define a quotient of an action by a group stack on a stack. In chapter 6, we define the notion of a 2-lien on a space X, following the approach of Giraud for an ordinary lien. We then do the necessary work of showing how to any 2-gerbe, one may asociate such a 2-lien, and that this association is given up to canonical 2-equivalence. After establishing this, we give a more down-to-earth description of the 2-lien of a 2-gerbe: it is an object that is given locally by a group stack, with descent data given up to inner equivalence. In chapter 7, our goal is to prove some theorems about G-2-gerbes, whose 2-lien arises from a strict Picard stack G. For this purpose, in section 1 we review Breen s treatment of the 3-cocycle description of a G-2-gerbe [4]. In section 2 of chapter 7, we prove results about G-2-gerbes which are analogues of theorems about for liens of gerbes. In section 3, we wish to give a nice cohomological description of connected G-2-gerbes where G is strict Picard. Deligne has shown that any strict Picard stack G corresponds to a 2-term complex of abelian sheaves K = [ K 0 d K 1 ]. We begin by reviewing Deligne s contruction, and then giving the definition of a degree 3 hypercohomology group. We then prove that the 10

11 set of equivalence classes Ȟ3 (X, G) of connected gerbes with 2-lien G is isomorphic to the hypercohomology group Ȟ3 (X, K ). We conclude with chapter 8, where we summarize our results, and indicate some directions for future work. 11

12 CHAPTER 2 GERBES, LIENS AND 2-GERBES 2.1 Torsors and H 1 We will rapidly review some elements of the theory of torsors/principal bundles [5], [7]. Recall that given sets X, Y and Z with maps f : X Z, g : Y Z, the fibered product X Z Y is the subset of the product X Y consisting of (x, y) such that f(x) = g(y). Let G be a sheaf of groups on a space X. Definition A right G-torsor on X is a space π : P X above X, together with a right group action P G P of G on P such that the induced morphism P X G P X P (2 1) (p, g) (p, pg) is an isomorphism. In addition, we require that there exist a family of local sections s i : U i P of π, for some open cover U = (U i ) i I of X. The trivial G-torsor is X G itself, with the action of G given by multiplication. Any G-torsor is locally isomorphic to the trivial torsor T G. A G-torsor is isomorphic to T G if and only if it has a global section s : X P. Now, given a space X and a sheaf G of groups ; we wish to consider isomorphism classes of G-torsors q : P X. Given an open covering (U i ) i I and a section s i of q over U i, we have a function g ij : U ij G Uij such that s j = s i g ij (recall that G acts on the right on P ). These transition functions satisfy the equality g ik = g ij g jk over U ijk. If the section s i is replaced over U i by s i = s i h i, then g ij is replaced by h 1 i g ij h j. This leads to the appropriate notion of 1-cocycles and coboundaries. For a sheaf G of (possibly nonabelian) groups on a space X, and (U i ) i I an open covering, we define a Čech 1-cocyle with values in G to consist of a family a ij Γ(U ij, G) such that the cocycle 12

13 relation a ik = a ij a jk (2 2) holds. Next, two Čech 1-cocycles a ij and a ij are said to be cohomologous if there exists a section h i of G over U i such that a ij = h 1 i a ij h j (2 3) thus defining an equivalence relation on the set of Čech 1-cocycles. Definition For a sheaf G of groups on the space X, and an open covering U = (U i ) i I of X, the first cohomology set Ȟ1 (U, G) is defined as the quotient of the set of 1-cocycles with values in G by the equivalence relation: a is cohomologous to ā. 2. The first cohomology set H 1 (X, G) is defined as the direct limit limȟ 1 (U, G) where U the limit is taken over the set of all open coverings of X, ordered by the relation of refinement. Note that the set H 1 (X, G) has a distinguished element, the class of the trivial 1-cocycle 1. So it is a pointed set. The definition of first cohomology makes the following result immediate. Proposition The set of isomorphism classes of G-torsors over X is in a natural bijection with the sheaf cohomology set H 1 (X, G). 2.2 Fibered Categories and Stacks We will review the definitions from category theory [4], [5] relevant to our discussion. Recall that a groupoid is a category in which every morphism is invertible. Definition A category fibered in groupoids above a space X consists of a family of groupoids C U, for each open set U in X, together with an inverse image functor f : C U C U1 (2 4) 13

14 associated to every inclusion of open sets f : U 1 U (which is the identity whenever f = 1 U ), and a natural transformation φ f,g : (fg) g f (2 5) for every pair of composable inclusions U 2 g U 1 f U. (2 6) For each triple of composable inclusions U 3 h U 2 g U 1 f U (2 7) we require that the composite natural transformations ψ f,g,h : (fgh) h (fg) h (g f ) (2 8) and χ f,g,h : (fgh) (gh) f (h g )f (2 9) coincide. We will frequently refer to the inverse image of an object x C U by an inclusion i : V U as the restriction x V of x above V. Definition A Cartesian functor F : C D between fibered categories consists of a family of functors F U : C U D U indexed by the open sets U, together with, for every morphism f : U 2 U 1, a natural transformation ϕ f : f F U1 F U2 f. (2 10) This is required to be compatible, for any pair of composable inclusions f and g, with the natural transformations (fg) g f for C and for D. Finally, a 2-arrow Ψ : F G between a pair of Cartesian functors F and G from C to D is defined 14

15 by a family of natural transformations Ψ U : F U G U which are compatible with the restriction functors f. Definition A prestack in groupoids above a space X is a fibered category in groupoids above X such that arrows glue i.e. for every pair of objects x, y C U, the presheaf Hom CU (x, y) is a sheaf on U. If, in addition, objects glue i.e. descent is effective for objects in C, then the prestack is called a stack in groupoids above X. A morphism of stacks is just a morphism of the underlying fibered categories. By descent data we mean that we are given, for every open cover U = (U α ) of an open set U X, a family of objects x α C Uα, and a family of isomorphisms φ αβ : x α Uαβ x β Uαβ such that φ αβ φ βγ = φ αγ (2 11) above U αβγ. The descent condition (x i, φ ij, ψ αβγ ) is effective if there exists an object x C U together with isomorphisms f α : x Uα x α compatible with the morphisms φ αβ i.e. the following diagram commutes: x Uαβ f α fβ x α Uαβ x β Uαβ φ αβ (2 12) By a construction based on the corresponding construction for sheaves, given any prestack C on X, there corresponds an associated stack C, together with a cartesian functor i : C C (2 13) which is universal for cartesian functors from C into stacks (see [18]). 15

16 Definition A stack in groupoids C on X, endowed with a monoidal associative Cartesian functor : C C C, with identity objects and with inverses will be called a group stack (or just a gr-stack) on X. For any V i Ob(C U ) (i=1, 2, 3, 4) the associativity isomorphisms α Vi V j V k : (V i V j ) V k V i (V j V k ) must sit within the following two commutative diagrams. 1. Pentagon Axiom. ((V 1 V 2 ) V 3 ) V 4 α 1,2,3 id 4 (V 1 (V 2 V 3 )) V 4 α 12,3,4 (V 1 V 2 ) (V 3 V 4 ) (2 14) α 1,23,4 V 1 ((V 2 V 3 ) V 4 ) α 1,2,34 id 1 α 2,3,4 V 1 (V 2 (V 3 V 4 )) 2. Triangle Axiom. α (V 1 1) V 2 ρ id id λ V 1 V 2 where ρ Vi : V i 1 V i and λ Vi : 1 V i V i. V 1 (1 V 2 ) (2 15) Definition A group category G is said to be braided when its group law is endowed with a commutivity isomorphism S X,Y : XY Y X which is functorial in X and Y and sits in a commutative square X1 m X S 1X X S (2 16) and two hexagonal associativity diagrams. G is said to be Picard if in addition S Y,X S X,Y = 1 XY for all X, Y in G. G is said to be a strict Picard category if in addition to all the above, S X,X = 1 X for all X. Definition A gr-stack is said to be a strict Picard stack if it is endowed with a commutativity natural transformation S for the group law S X,Y and satisfies the corresponding conditions. 16

17 2.3 Gerbes and their Liens The following definitions are due to Giraud [15]. We follow the presentation in [4], [5] and [20]. To say that a stack G is locally non-empty means there exists a covering U = (U i ) of X for which the set of objects of the category G Ui is non-empty. The locally connectedness condition on G is the requirement that for any pair of objects x and y in G U, there exists an open cover V = (V α ) of U such that the set of arrows from x Vα to y Vα is non-empty for all α. Definition A (1)-gerbe on a space X is a stack in groupoids G on X which is locally non-empty and locally connected. A morphism of gerbes is just a morphism of the underlying fibered categories. A gerbe G on X is said to be neutral (or trivial) when the fiber category G X is non-empty. A lien on a space X, as first defined by Giraud in his thesis [15], is a collection (G i ) of sheaves of groups corresponding to an open cover (U i ) of X with descent data up to inner conjugation. In other words a lien on X is an object which is defined locally by a sheaf of groups, but in a category where morphisms between groups differing by inner conjugation are identified. We give a cocycle description of a lien. Let U = (U i ) ı I be an open cover of X, and suppose we have a family of sheaves G i of groups, defined on the open sets U i. Denote each sheaf by lien(g i ). The sheaves lien(g i ) and lien(g j ) are glued on the open set U ij by a section ψ ij of the quotient sheaf Out(G j, G i ) := Isom(G j, G i )/Int(G i ) on U ij. Thus the lien L is determined by a family of sections of Out(G j, G i ) on U ij satisfying the 1-cocycle condition ψ ij ψ jk = ψ ik in Out(G k, G i ) and the normalization condition ψ ii = 1 for all i. 17

18 Two liens L and L on X, each defined locally by families of groups (G α ) and (G β ), are isomorphic whenever there exists a common refinement V = (V i ) of the defining covers U and U, and a family of isomorphisms χ i : lien(g i ) lien(g i) on the open sets V i, which are compatible with the gluing data. The cocycles (ψ ij ) and (ψ ij ) are therefore related by the coboundary conditions ψ ij = χ i ψ ij χ 1 j, the isomorphisms χ i are viewed as sections on the open sets V i of the sheaf Out(G i, G i ) of outer automorphisms of G i. Let G be a sheaf of groups, and let G i = G Ui. Then when a lien is locally of the form lien(g i ), each ψ ij is a section on the set U ij of the sheaf Out(G) of outer automorphisms of G. Following [4] we will call such liens, which are locally isomorphic to the lien lien(g), G-liens. It follows from this description that they are classified by the non-abelian cohomology set H 1 (X, Out(G)). Let G be a gerbe. It is locally non-empty so there exists an open cover (U α ) of X such that G(U α ) is non-empty. So choose a family of objects x α G(U α ). Now Aut(x α ) is a sheaf on U α. Now G is also locally connected so for each α, β there exists an open cover of (U ξ αβ ) of U αβ and for each ξ an arrow f ξ αβ : x β x α in G(U ξ αβ ). Conjugation by this arrow defines an isomorphism of sheaves of groups λ ξ αβ : Aut(x β) U ξ αβ This isomorphism λ ξ αβ depends on the choice of the f ξ αβ Aut(x α ) U ξ. αβ but different choices define the same outer isomorphism. In particular, on overlaps U ξ αβ U ζ αβ the two isomorphisms λξ αβ and λ ζ αβ define the same outer isomorphism. Thus for fixed α, β the family λξ αβ defines an outer isomorphism of sheaves on U αβ : λ αβ : Aut(x β ) Uαβ Aut(x α ) Uαβ (2 17) which does not depend on the choice of f ξ αβ. This system of sheaves of groups Aut(x α) and outer isomorphisms λ αβ is called the lien of the gerbe G. To any gerbe with lien L, we can attach a 2-cocycle that takes its values in L, see [20] for details. Definition Let K be a lien on X. A gerbe with lien K is a gerbe G on X together with an isomorphism of liens θ : lien(g) K. Two gerbes with lien K are said to be 18

19 equivalent if there is an equivalence between the underlying fibered categories. We designate by H 2 (X, K) the set of equivalence classes of gerbes with lien K. Example Let G be a bundle of groups on X. The stack T ors(g) of right G-torsors on X is a gerbe on X. It is (globally) non-empty since its fiber on X always contains the trivial G-torsor on X. It is also locally connected since any G-torsor is locally isomorphic to the trivial G-torsor. Thus T ors(g) is in fact a neutral gerbe. Its lien is the lien represented by the group G, denoted lien(g). Example Let 1 A C B 1 be an exact sequence of (possibly infinite dimensional) Lie groups such that the projection C B is a locally trivial A-bundle. Let p : P X be a smooth B-bundle over a manifold X. Consider the problem of finding a C-bundle q : Q X such that the associated B-bundle Q/A X is isomorphic to P X. The fibered category of local solutions to this is a gerbe on X. The lien associated to this gerbe is isomorphic to the sheaf A X of smooth A-valued functions. Definition Let P be a gerbe on X, let G be a sheaf of groups on X, and let U = (U i ) i I be an open cover of X. We say that P is a G-gerbe on X if there exists for each i I an object x i Ob(P Ui ) and an isomorphism of sheaves of groups on U i η i : G Ui Aut PUi (x i ) (2 18) where Aut PUi (x i ) is the sheaf of automorphisms of the object x i above U i. Definition Let P be a gerbe on X, let G be a sheaf of groups on X. Suppose there exists, for every object x in a fiber category P U, an isomorphism of sheaves of groups η x : G U Aut PU (x), (2 19) and that, for any morphism f : x y in P U, the corresponding diagram of sheaves on U η x Aut PU (x) G U λ η y Aut PU (y) (2 20) 19

20 determined by the morphism λ associated to f commutes. The gerbe P is then called an abelian G-gerbe on X. The map λ associated to f is the isomorphism λ : Aut PU (x) Aut PU (y) (2 21) u fuf 1. (2 22) An abelian G-gerbe is a G-gerbe by definition. In fact, we now show that in this situation the sheaf G must be abelian. In particular, the commutivity of the group law in G can be verified locally, for sections of a sheaf G U. Let g be a section of this sheaf and consider the above triangle associated to the corresponding arrow u = η x (g) : x x. This is the diagram η x Aut PU (x) G U η x Aut i PU (x) u where i u denotes inner conjugation by u in the sheaf Aut PU (x). Commutivity of this (2 23) diagram implies that i u is the identity map whence the sheaf Aut PU (x), and therefore G U is abelian. We state two propositions (see [4] for proofs) to illustrate how the notion of a lien is helpful in characterizing a G-gerbe. Proposition Let G be a sheaf of groups on a space X. A gerbe G on X is a G-gerbe if and only if its lien is locally isomorphic to lien(g). Proposition Let G be a sheaf of abelian groups on a space X. A gerbe G on X is an abelian G-gerbe if and only if lien(g) lien(g). We now state a theorem of Giraud (see [15]) giving the connection between gerbes with G-liens where G is an abelian sheaf, and cohomology. Theorem Let G be a sheaf of abelian groups on X. Let L = lien(g). A gerbe G with lien L gives rise to a degree-2 Čech cocycle with values in L as follows. Choose an 20

21 open covering (U i ) of X for which each G(U i ) is nonempty, and such that any two objects of G(U ij ) are isomorphic. Choose objects P i of G(U i ), and let φ ij : P j Uij P i Uij be an isomorphism between objects in the category G(U ij ). We define a section h ijk of L over U ijk by φ ijk = φ 1 ik φ ij φ jk Aut(P k ) L (2 24) so that (h) = (h ijk ) is an L-valued Čech 2-cocycle. The corresponding class in H2 (X, L) is independent of all choices, and the assignment G (h) gives a one-to-one correspondence between equivalence classes of gerbes with lien L and the elements of H 2 (X, L). We conclude this section with the following definition. Definition Let G be a gr-stack on X. A right G-torsor on X is a stack Q on X together with a right action functor Q G Q (2 25) which is coherently associative and satisfies the unit condition, and for which the induced functor Q G Q Q (2 26) (q, g) (q, qg) (2 27) is an equivalence. In addition we require that Q be locally non-empty Categories, 2-Functors, 2-Natural Transformations We will review, in an informal spirit, some definitions from the theory of 2-categories. Definition A 2-category A consists of the following data: 1. a collection of objects A, B, C,..., 2. for each ordered pair of objects (A, B), a small category A(A, B), 3. for each triple A, B, C of objects, a bifunctor c A,B,C : A(A, B) A(B, C) A(A, C), (2 28) 21

22 4. for each object A, a functor u A : 1 A(A, A). (2 29) Here 1 denotes the terminal category (with one object and one arrow) in the category of small categories. These elements of data are required to satisfy associativity laws for composition and the requirement that u A provides a left and right identity for this composition. Example A basic example is the 2-category CAT, whose objects are small categories, arrows are functors, and 2-arrows are natural transformations. Definition Given two 2-categories A and B, a 2-functor F : A B consists in giving 1. for each object A A, an object F (A) B, 2. for each pair of objects A, A in A, a functor F A,A : A(A, A ) B(F (A), F (A )) (2 30) (For brevity s sake we often write F instead of F A,A ). This data is required to satisfy the following axioms: 1. Compatibility with composition: given three objects A, A, A in A, the following diagram commutes. A(A, A ) A(A, A ) c AA A A(A, A ). (2 31) F AA F A A F AA B(F (A), F (A )) B(F (A ), F (A )) c F (A)F (A )F (A ) B(F (A), F (A )) 22

23 2. Unit: for every object A A the following diagram commutes. u A 1 A(A, A) (2 32) F AA u F (A) B(F (A), F (A)) Definition Consider two 2-categories A, B, and two 2-functors A F B between them. A 2-natural transformation θ : F G consists in giving, for each object A A, an arrow θ A : F (A) G(A) such that the following diagram commutes for each pair of objects A, A. G A(A, A ) F AA B(F (A), F (A )). (2 33) G AA B(1 F (A),θ A ) B(G(A), G(A )) B(θ A,1 G(A ) ) B(F (A), F (A )) 2.5 Fibered 2-Categories, 2-Stacks and 2-Gerbes Definition A fibered 2-category in 2-groupoids above a space X consists of a family of 2-groupoids C U, for each open set U in X, together with an inverse image 2-functor f : C U C U1 (2 34) associated to every inclusion of open sets f : U 1 U (which is the identity whenever f = 1 U ), and a natural transformation φ f,g : (fg) g f (2 35) for every pair of composable inclusions U 2 g U 1 f U. (2 36) 23

24 For each triple of composable inclusions U 3 h U 2 g U 1 f U (2 37) we require a modification (which is a map between natural transformations) ψ f,g,h (fgh) α f,g,h h g f (2 38) χ f,g,h between the composite natural transformations ψ f,g,h : (fgh) h (fg) h (g f ) (2 39) and χ f,g,h : (fgh) (gh) f (h g )f. (2 40) Finally, for any U 4 U 3, the two methods by which the modifications α compare the composite two arrows (fghk) (ghk) (f) ((hk) g )f k h g f (2 41) and must coincide. (fghk) k (fgh) k (h (fg) ) k h g f (2 42) Definition A Cartesian 2-functor F : C D between fibered 2-categories over X consists of a family of 2-functors F U : C U D U indexed by the open sets U, together with, for every morphism f : U 2 U 1, a natural transformation of 2-functors φ f : f F U1 F U2 f. (2 43) For any pair of composable inclusions f and g, a 3-arrow α f,g is also given, which compares the natural transformation from (fg) F U1 to F U3 g f defined by φ f and φ g with 24

25 that constructed from φ fg. This 3-arrow is required to satisfy a coherence condition relating the two induced natural transformations associated to a triplet (f, g, h) of composable inclusions. This condition will not be made explicit here. Definition A 2-prestack in groupoids C above a space X is a fibered 2- category in 2-groupoids above X such that for every pair of objects x, y C U, the fibered category Ar CU (x, y) is a stack on U. If, in addition, 2-descent is effective for objects in C, then the 2-prestack is called a 2-stack in 2-groupoids above X. A morphism of 2-stacks is just a morphism of the underlying fibered 2-categories. By 2-descent data we mean that we are given, for an open cover U = (U α ) of an open set U X, a family of objects x i C Ui, of 1-arrows φ αβ : x α Uαβ x β Uαβ and a family of 2-arrows ψ αβγ : φ αγ φ βγ φ αβ x β φ βγ φ αβ ψ x γ φ αγ x α (2 44) for which the tetrahedral diagram of 2-arrows induced by ψ in C Uαβγδ commutes: x δ (2 45) x γ x α x β The 2-descent condition (x i, φ ij, ψ αβγ ) is effective if there exists an object x C U together with isomorphisms f α : x Uα x α compatible with the morphisms φ ij and ψ αβγ. For any 2-prestack C on X, one defines by the same sheafification method as for sheaves and 1-stacks, an associated 2-stack 2-functor a : C C (2 46) 25

26 which is universal for Cartesian 2-functors b : C D from C into 2-stacks. The latter property characterizes C upto 2-equivalence. For, suppose that a Cartesian 2-functor b : C D satisfies: 1. b is fiberwise fully faithful.i.e. for any pair of objects x and y in a fiber category C U, the induced map of stacks on U Ar(x, y) Ar(b(x), b(y)) (2 47) is an equivalence. 2. every object in D is locally isomorphic to one in the image of C. Then D is an associated 2-stack of C. Definition A 2-gerbe P on a space X is a 2-stack in 2-groupoids on X which is locally non-empty, locally connected, in which 1-arrows are weakly invertible, and 2-arrows are invertible. A morphism of 2-gerbes is just a morphism of the underlying 2-stacks. To say that P is locally non-empty means there exists a covering U = (U i ) of X for which the set of objects of the 2-category P Ui is non-empty. The connectedness condition on P is the requirement that for any pair of objects x and y in the fibered 2-category G U, there exists an open cover V = (V α ) of U such that the set of 1-arrows from x Vα to y Vα is non-empty for all α. To say that 1-arrows are weakly invertible means that for any 1-arrow f : x y in a fibered 2-category P U, there exists an arrow g : y x in P U which is both a left and right inverse of f (upto a pair of 2-arrows λ : g f 1 x and ρ : f g 1 y ). Example (Breen) Let L be a lien on X. We can ask when is L isomorphic to a lien of the form lien(g) for some gerbe G on X? Locally the answer is always, since the lien L is locally isomorphic to a lien of the form lien(g) for some sheaf of groups G, whence it is realized by the neutral gerbe T ors(g) corresponding to G. Globally this gives a 2-gerbe on X. 26

27 For a 2-gerbe P, we can associate to each object x P U a gr-stack P x := Ar U (x, x) above U. Definition Let P be a 2-gerbe on X, and let G be a gr-stack on X. We say that P is a G-2-gerbe on X if there exists an open cover U = (U i ) i I of X, for each i I an object x i Ob(P Ui ) and an equivalence G Ui G xi over U i. Definition Let P be a 2-gerbe on X, let G be a gr-stack on X. Suppose there exists, for every object x in a fiber 2-category P U, an isomorphism of sheaves of gr-stacks η x : G U Eq PU (x), (2 48) and for any morphism f : x y in P U, a 2-arrow η f : λ f η x η y η x Eq PU (x) G U η y η f Eq λ PU (y) (2 49) where λ f is the morphism of gr-stacks λ : Eq PU (x) Eq PU (y) (2 50) u fuf 1. (2 51) defined by f. The natural transformations η f are required to respect the group structures, and satisfy the following transitivity and normalization conditions: 1. For any pair of composable morphisms f : x y and g : y z in P U, the composite 2-arrow obtained by pasting η f and η g is equal to η gf. 2. For any 2-arrow φ : f g between a pair of morphisms f, g : x y in P U, η f = η g λ φ, where λ φ : λ f λ g is conjugation by φ. 3. For every x in P U, η 1x = 1 A 2-gerbe P satisfying these conditions is called an abelian G-2-gerbe on X. 27

28 2.6 Representable Functors We will rapidly review the notion of a representable functor. The primary source for this summary is [12]. Let C be a category and let SET denote the category of sets. The category of presheaves of sets on C is the functor category Hom(C op, SET ). Now choose an object X of C and define a presheaf h X by: h X (Y ) = Hom C (Y, X), h X (f) = Hom C (f, X) := f (2 52) for any object Y of C and arrow f in C. This functor is called the functor represented by X. This construction is functorial in X, i.e. gives a functor h : C Hom(C op, SET ) (2 53) defined by h(x) = h X, and for f Hom(X, Y ), h(f)(z) : Hom(Z, X) Hom(Z, Y ) (2 54) g f g. (2 55) Definition A functor F Hom(C op, SET ) is representable if it is in the essential image of h, i.e. if it is isomorphic to some h X. We say that F is represented by (the object) X. Note that the isomorphism is part of the data. If (X, α : h X F ) represents F then we obtain a universal element ξ = α X (id X ) F (X). The isomorphism can be recovered from ξ. To see this, if f : X Y is any arrow in C, then the diagram Hom(X, X) α X F (X). (2 56) Hom(f,X) Hom(Y, X) F (f) α Y F (Y ) 28

29 commutes, since α is a natural transformation. But id X Hom(X, X) maps under Hom(f, X) to f Hom(Y, X), whence by the above diagram we have α Y (f) = F (f)(α X (id X )) = F (f)(ξ) (2 57) which shows how to compute α from ξ. Thus we can also say F is represented by the pair (X, ξ). Yoneda s lemma states that: Theorem The functor h : C Hom(C op, SET ) is fully faithful. Corollary If a functor F is representable then the representing object in C is defined uniquely up to a unique isomorphism. In the category SET we have definitions for products, fibered products, equalizers, coequalizers, etc. We can use representable functors to import these notions from SET to any arbitrary category. To see how this is done we will examine carefully the definition of product. Products. The product of two sets X and Y, together with the projection maps p 1 : X Y X, p 2 : X Y Y, is characterized up to isomorphism by a universal property: given any sets Z with maps q 1 : Z X, q 2 : Z Y, there is a unique map q : Z X Y such that q 1 = p 1 q and q 2 = p 2 q. In fact, if P and P are two sets with this universal property, then there exist maps f : P Q, g : Q P and uniqueness implies g f = id P and f g = id Q. Thus the product of two sets can be defined as the object with this universal property. This description of the product of the sets X and Y mentions only objects and morphisms; it says nothing about ordered pairs or any thing specifically about the objects under consideration. So we can import this definition to any arbitrary category. In particular, if C is any random category and X, Y are objects of C, we can define the product of X and Y as being an object P of C with morphisms p 1 : P X, p 2 : P Y which is universal, i.e. if Z is any other object of C with maps q 1 : Z X, q 2 : Z Y, there is a unique map q : Z P such that q 1 = p 1 q and 29

30 q 2 = p 2 q. In general such a universal object may not exist. In the category SET it does: it is the set consisting of the ordered pairs (x, y) for x X, y Y. Universal properties (such as products) can be expressed by saying that a certain presheaf is representable. Definition If for every pair of objects X and Y of a category C, the presheaf h X h Y : Z Hom(Z, X) Hom(Z, Y ) (2 58) is representable, then we say that products are representable in C. The object that represents this functor is unique up to unique isomorphism whence we say that the product of X and Y is the object that represents this functor. To see why this definition makes sense, say an object P represents the product of X and Y. By definition this means we have an isomorphism h P h X h Y, i.e. an isomorphism α Z : Hom(Z, P ) Hom(Z, X) Hom(Z, Y ) (2 59) for each Z Ob(C). Then plugging P in for Z above, we get that the universal element α Z (id P ) is a pair of morphisms p 1 : P X, p 2 : P Y with the property that given any pair of morphisms q 1 : Z X, q 2 : Z Y, both factor through a unique morphism Z P, see diagram below. q Hom(Z, P ) Hom(Z, X) Hom(Z, Y ) (2 60) Hom(q,P ) (q 1 =p 1 q,q 2 =p 2 q) id P Hom(P, P ) Hom(P, X) Hom(P, Y ) The definition for arbitrary products is an obvious extension of the above definition. We can make similar definitions for fibered products, equalizers, and coequalizers. First we list these definitions for the category SET. Given sets X, Y and Z with maps 30

31 f : X Z, g : Y Z, the fibered product X Z Y is the subset of the product X Y consisting of (x, y) such that f(x) = g(y). Given two maps f, g : X Y of sets, the equalizer Eq(f, g) of f and g is the set {x X f(x) = g(x)}. If i : Eq(f, g) X is the inclusion, then f i = g i, and Eq(f, g) is universal for this property. The definition of the coequalizer is just dual to that of the equalizer. We now make this definitions for any arbitrary category C. Definition Given arrows f : X Z, g : Y Z in C, we define the fibered product X Z Y of X and Y to be the object P which represents the functor Q Hom(Q, X) Hom(Q,Z) Hom(Q, Y ). (2 61) If the object P exists then we have a functorial isomorphism α Q : Hom(Q, P ) Hom(Q, X) Hom(Q,Z) Hom(Q, Y ). (2 62) The universal object α P (id P ) is a pair of morphisms p 1 : P X, p 2 : P Y. Since it is universal, for any pair of morphisms q 1 : Q X, q 2 : Q Y such that f q 1 = g q 2, there is a unique arrow h : Q P such that q 1 = p 1 f and q 2 = p 2 f. 2. Given arrows f, g : X Y, we define their equalizer to be the object K which represents the functor Z Eq( Z X f g Y ). (2 63) i.e., Z Eq( Hom(Z, X) Hom(Z,f) Hom(Z,g) If K exists then there is a functorial isomorphism Hom(Z, Y ) ). (2 64) Hom(Z, K) Eq(Hom(Z, f), Hom(Z, g)). (2 65) 31

32 Thus there is a canonical morphism i : K X such that f i = g i, and any morphism h : Z X such that f h = g h factors through a unique morphism Z K. 3. Given arrows f, g : X Y, we define their coequalizer to be the object K which represents the functor Z Eq( Hom(X, Z) Hom(f,Z) Hom(g,Z) Hom(Y, Z) ). (2 66) If K exists then there is a functorial isomorphism Hom(Z, K) Eq(Hom(f, Z), Hom(g, Z)). (2 67) Thus there is a canonical map p : Y K such that p f = p g, and any morphism h : Y Z such that h f = h g factors through a unique morphism K Z Representability We will now briefly discuss the notion of 2-representability. These definitions can be found in [16]. Let C be a 2-category. Recall that C op is obtained from C by inverting the direction of the 1-cells of C.i.e. if [X, Y ] is the collection of 1-cells in C from X to Y then [X, Y ] in C op is [Y, X]. We will define 2-representability for 2-functors f : C op CAT. Let F (C) be the 2-category of 2-functors from C op to CAT. We define the following strict 2-functor: h : C CAT (2 68) h X ( ) = Hom C (, X) (2 69) and which is defined similarly on 1-cells and 2-cells. Let f be any 2-functor.i.e. f Ob(F (C)). Then for all X Ob(C), we can define the functor h X, : Hom F (C) (h X, f) f(x). (2 70) 32

33 For u Ob(Hom F (C) (h X, f)) we associate h X,f (u) = u(id X ) Ob(f(X)) and to φ : u u, where φ Mor(Hom F (C) (h X, f)) we associate h X,f (φ) = φ(id X ) : u(id X ) u (id X ). (2 71) The functor h X,f is an equivalence of categories with the quasi-inverse given by k X,f : f(x) Hom F (C) (h X, f) where for all ξ Ob(f(X)), k X,f (ξ) = u ξ is the morphism of 2-functors u ξ : h X f, u ξ ( ) = f( )(ξ) and for φ : ξ ξ Mor(f(X)), k X,f (φ) = f( )(φ) : u ξ u ξ. (2 72) In particular, letting f = h X we get the following proposition. Proposition The 2-functor h : C F (C) is 2-faithful. Definition We say that a 2-functor f Ob(F (C) is 2-representable if f is equivalent to a 2-functor of the form h X. We say that the solution to the universal 2- problem determined by f is the pair (X, u) where X Ob(C) and u : h X f is an equivalence of 2-functors. We say that the pair (X, ξ), where ξ = u(id X ) Ob(f(X)) gives a representation of f. See [16] for the proofs of the following propositions. Proposition Let f be a representable 2-functor in F (C) and let (X, ξ) and (X, ξ ) be two representations of f. Then there exists a pair (λ, ɛ) which gives an equivalence in C: λ : X X and a 2-isomorphism ɛ : f(λ)(ξ ) ξ. Further, for all other such pairs (λ 1, ɛ 1 ) there exists a unique 2-isomorphism α : λ λ 1 such that ɛ 1 (f(λ)(ξ )) = ɛ. Proposition Let C be a 2-category and let F(C) denote the 2-category of 2- functors from C op to CAT. Let F be a 2-functor which is representable in F(C), and suppose (X, ξ) and (X, ξ ) are two given representations of F. Then there exists a pair (λ, ɛ) such that λ : X X is an equivalence in C and ɛ : F (λ)(ξ ) ξ is a 2-isomorphism. Further, for every other such pair (λ 1, ɛ 1 ) there exists a unique 2-isomorphism α : λ λ such that ɛ 1 (F (λ)(ξ )) = ɛ. 33

34 2.8 Giraud s approach to Liens of Gerbes We now describe Giraud s original definition of a lien given in [15]. It is this approach that we will generalize to define the 2-lien of a 2-gerbe. Let X be a topological space, and let F and G be sheaves of groups on X. Consider the sheaf Isom(F, G) (the sheaf of isomorphisms of sheaves of groups from F to G). Let Int(G) denote the sheaf of inner automorphisms of the sheaf G. Define the quotient sheaf Out(F, G) := Isom(F, G)/Int(G) (2 73) where Int(G) acts on Isom(F, G) from the right via composition. Now, for every open U X, let LI(X)(U) denote the category whose objects are the sheaves of groups on U, and whose arrows between two objects A and B are the global sections of the sheaf Out(A, B)(U) i.e. Hom LI(X)(U) (A, B) = Γ(U, Out(A, B)). The composition law is defined after passing to the quotient sheaf Out, and is well-defined (see the discussion of a lien of a gerbe in Section 2.3). Now if V U is an inclusion of open sets in X, then the formation of the quotient Out(A, B) commutes with the restriction of U to V. So we can define a fibered category LI(X). Let Grp(X) denote the fibered category of sheaves of groups on X. Then by construction we have a morphism of fibered categories Grp(X) LI(X). (2 74) Now if A and B are any two objects of the category LI(X)(U), then the presheaf Hom LI(X)(U) (A, B) of arrows from A to B is identified with the sheaf Out(A, B). Thus LI(X) is a prestack. Thus upon applying the stackification functor, we obtain the associated stack LIEN(X). Composing the stackification functor [15] with the morphism (1) yields a morphism of stacks lien(x) : Grp(X) LIEN(X). (2 75) 34

35 Definition The stack LIEN(X) is called the stack of liens on X. 2. We call a global section of this stack a lien over X. For every U X, we say that a lien of U is an object of the category LIEN(X U ) which is the fiber over U of the stack of liens. We call LIEN(X U ) the category of liens over U. To generalize this definition of a lien from topological spaces to topoi, we just replace global section in Definition with a cartesian section of the stack LIEN(X), as done by Giraud in [15]. Now let G be a stack on X. Recall that Grp(X) denotes the stack of sheaves of groups on X. We have a cartesian functor AUT : G Grp(X) (2 76) AUT (G)(x) = Aut U (x) (2 77) where x Ob(G(U)). Composing this morphism with the morphism lien(x) : Grp(X) LIEN(X) gives a morphism of stacks liau(g) : G LIEN(X) (2 78) which to every object x G(U) associates the lien liau(g)(x) = lien(aut U (x)) (2 79) (called the lien represented by the sheaf of U-automorphisms of x) and to every U-isomorphism α : x y of G associates the morphism liau(g)(α) = lien(int(α)) (2 80) (where Int(α) : Aut U (x) Aut U (y) defined by Int(α)(a) = αaα 1 is a morphism of sheaves of groups over X). 35

36 By the definition of the stack of liens, if m, n : x y are two U-isomorphisms of G, we have liau(g)(m) = liau(g)(n) (2 81) since the functor lien transforms every inner automorphism to an identity arrow. Now let m : F G be a morphism of stacks. Then we have a morphism of morphisms of stacks (i.e a 2-arrow): liau(m) : liau(f) liau(g) m (2 82) where, liau(m) = lien(x) AUT (m). (2 83) For every x Ob(F(U)), the functor m induces a morphism of sheaves of groups µ x : Aut U (x) Aut U (m(x)) (2 84) and by definition of the map liau(m), the morphism liau(m(x)) is the morphism of liens represented by µ x : m x = lien(µ x ), m x : lien(aut U (x)) lien(aut U (m(x))). (2 85) Definition Let F be a stack over X and let L be a lien over X i.e. it is a global section L : X LIEN(X) of the stack of liens. An action of L on F is a morphism of morphisms of stacks: a : L f liau(f) (2 86) where f : F X is the projection onto the Zariski site of X F f X liau(f) L a LIEN(X) (2 87) 36

37 The Zariski site of X is the category whose objects are the open sets of X, and the morphisms are the inclusion maps. By definition a is a morphism of (cartesian) functors i.e. a family a(x) : L(U) lien(aut U (x)), x Ob(F(U)), U X, of morphisms of liens on U that satisfies the conditions of compatibility with the restriction to open sets, and with the composition of morphisms. Definition Let (L, a) be a lien operating on a stack F and let u : L L be a morphism of liens over X. Then the action induced by a and u is the morphism b defined by b : L f liau(f ), (2 88) b = a (u f L) (2 89) (where f : F X is the projection onto the Zariski site of X). So b is the action such that, for every x Ob(F(U)), b(x) is the composition L (U) u(u) L(U) a(x) liau(f)(x). (2 90) Proposition Let F be a gerbe on X. 1. Let (L, a) be a lien operating on F. The following are equivalent: (a) a is an isomorphism. (b) for every lien L on X and every action b of L on F there exists a unique morphism of liens u : L L such that b is the action induced by a and u. 2. There exists a lien (L, a) operating on F that satisfies the conditions of (1). Definition We say that the lien of a gerbe F is a lien operating on F and satisfying the conditions of the above proposition. Condition 1(b) characterizes the lien of a gerbe F up to canonical equivalence. So we are justified in saying the lien of a gerbe F. By abuse of notation, the map a is often not mentioned. If L is the lien of a gerbe F, we say F is bound by L. Conversely, if L is a lien on X, then an L-gerbe is a pair (F, a) where F is a gerbe and (L, a) is a lien of F. 37

38 The lien of a gerbe F was in fact made explicit before: it is a lien L and a family of isomorphisms of liens over U, a(x) : L(U) lien(aut U (x)) (2 91) where x F(U), U X. This family is required to be compatible with the restriction of open sets and also satisfy the condition that if i : x y is a U-isomorphism of F, then the morphism of sheaves of groups Int(i) : Aut U (x) Aut U (y) represents the identity morphism of L(U). Proof of Proposition We follow Giraud s proof in [15]. Since F is a gerbe, the projection f : F X zar to the Zariski site is fully faithful, hence the map Hom(L, L) Hom(L f, Lf) given by u u f is bijective. This forces b = a(u f) whence we have that 1(a) implies 1(b). Next, we show that for every gerbe F, there exists a lien L and an isomorphism a : Lf liau(f). Then (2) follows trivially, and so does 1(b) 1(a) since 1(b) determines L upto unique isomorphism. Let I be the image category of liau(f). Clearly this is a fibered category of LIEN(X) and the functor induced by liau(f), L : F I (2 92) is cartesian. Moreover, the projection I X zar is both fully faithful and essentially surjective (this follows trivially from the fact that F X is fully faithful and because liau(m) = liau(n) for any two U-isomorphisms m, n : x y of F). By the universal property of the associated stack, it follows that X is the stack associated to I, and that there exists a cartesian section L 0 : X I of I and an isomorphism a 0 : L 0 f L. Then setting L = il 0 and a = i a 0 (where i : I LIEN(X) is the inclusion) the conclusion follows. Corollary Let m : F G be a morphism of gerbes and let (L, a) and (M, b) denote the liens of F and G respectively. Then there exists a unique morphism of liens u : L M such that m is a u-morphism i.e. liau(m) a = (b m)(u f). 38

39 CHAPTER 3 EQUALIZERS AND COEQUALIZERS Recall that a lien is defined by assigning a family of sheaves of groups (G i ) to the open sets (U i ) that cover X, and these sheaves are glued on the overlaps U ij by a section of the quotient sheaf Out(G j, G i ) = Isom(G j, G i )/Inn(G i ). Our project involves appropriately defining the action of a gr-stack on a stack, and the quotient by such an action. To do this we first need to define these notions for group categories. 3.1 Equalizers We begin by recalling the definition of equalizer for the category SET. Given two maps f, g : X Y of sets, the equalizer Eq(f, g) of f and g is the set {x X f(x) = g(x)}. For any arbitrary category C, given arrows f, g : X Y, we define their equalizer to be the object K which represents the functor Z Eq( Z X f g Y ). (3 1) i.e., Z Eq( Hom(Z, X) Hom(Z,f) Hom(Z,g) If K exists then there is a functorial isomorphism Hom(Z, Y ) ). Hom(Z, K) Eq(Hom(Z, f), Hom(Z, g)). (3 2) Thus there is a canonical morphism i : K X such that f i = g i, and any morphism h : Z X such that f h = g h factors through a unique morphism Z K. Equalizers are representable in the category SET by the object Eq(f, g). In fact we have a functorial isomorphism Hom(Z, Eq(f, g)) Eq(Hom(Z, f), Hom(Z, g)). (3 3) 39